sysdeps/ieee754/ldbl-128ibm/k_tanl.c

   1 /*
   2  * ====================================================
   3  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
   4  *
   5  * Developed at SunPro, a Sun Microsystems, Inc. business.
   6  * Permission to use, copy, modify, and distribute this
   7  * software is freely granted, provided that this notice
   8  * is preserved.
   9  * ====================================================
  10  */
  11
  12 /*
  13   Long double expansions are
  14   Copyright (C) 2001 Stephen L. Moshier <moshier@na-net.ornl.gov>
  15   and are incorporated herein by permission of the author.  The author
  16   reserves the right to distribute this material elsewhere under different
  17   copying permissions.  These modifications are distributed here under
  18   the following terms:
  19
  20     This library is free software; you can redistribute it and/or
  21     modify it under the terms of the GNU Lesser General Public
  22     License as published by the Free Software Foundation; either
  23     version 2.1 of the License, or (at your option) any later version.
  24
  25     This library is distributed in the hope that it will be useful,
  26     but WITHOUT ANY WARRANTY; without even the implied warranty of
  27     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
  28     Lesser General Public License for more details.
  29
  30     You should have received a copy of the GNU Lesser General Public
  31     License along with this library; if not, see
  32     <http://www.gnu.org/licenses/>.  */
  33
  34 /* __kernel_tanl( x, y, k )
  35  * kernel tan function on [-pi/4, pi/4], pi/4 ~ 0.7854
  36  * Input x is assumed to be bounded by ~pi/4 in magnitude.
  37  * Input y is the tail of x.
  38  * Input k indicates whether tan (if k=1) or
  39  * -1/tan (if k= -1) is returned.
  40  *
  41  * Algorithm
  42  *      1. Since tan(-x) = -tan(x), we need only to consider positive x.
  43  *      2. if x < 2^-57, return x with inexact if x!=0.
  44  *      3. tan(x) is approximated by a rational form x + x^3 / 3 + x^5 R(x^2)
  45  *          on [0,0.67433].
  46  *
  47  *         Note: tan(x+y) = tan(x) + tan'(x)*y
  48  *                        ~ tan(x) + (1+x*x)*y
  49  *         Therefore, for better accuracy in computing tan(x+y), let
  50  *              r = x^3 * R(x^2)
  51  *         then
  52  *              tan(x+y) = x + (x^3 / 3 + (x^2 *(r+y)+y))
  53  *
  54  *      4. For x in [0.67433,pi/4],  let y = pi/4 - x, then
  55  *              tan(x) = tan(pi/4-y) = (1-tan(y))/(1+tan(y))
  56  *                     = 1 - 2*(tan(y) - (tan(y)^2)/(1+tan(y)))
  57  */
  58
  59 #include <float.h>
  60 #include <libc-internal.h>
  61 #include <math.h>
  62 #include <math_private.h>
  63 static const long double
  64   one = 1.0L,
  65   pio4hi = 7.8539816339744830961566084581987569936977E-1L,
  66   pio4lo = 2.1679525325309452561992610065108379921906E-35L,
  67
  68   /* tan x = x + x^3 / 3 + x^5 T(x^2)/U(x^2)
  69      0 <= x <= 0.6743316650390625
  70      Peak relative error 8.0e-36  */
  71  TH =  3.333333333333333333333333333333333333333E-1L,
  72  T0 = -1.813014711743583437742363284336855889393E7L,
  73  T1 =  1.320767960008972224312740075083259247618E6L,
  74  T2 = -2.626775478255838182468651821863299023956E4L,
  75  T3 =  1.764573356488504935415411383687150199315E2L,
  76  T4 = -3.333267763822178690794678978979803526092E-1L,
  77
  78  U0 = -1.359761033807687578306772463253710042010E8L,
  79  U1 =  6.494370630656893175666729313065113194784E7L,
  80  U2 = -4.180787672237927475505536849168729386782E6L,
  81  U3 =  8.031643765106170040139966622980914621521E4L,
  82  U4 = -5.323131271912475695157127875560667378597E2L;
  83   /* 1.000000000000000000000000000000000000000E0 */
  84
  85
  86 long double
  87 __kernel_tanl (long double x, long double y, int iy)
  88 {
  89   long double z, r, v, w, s;
  90   int32_t ix, sign, hx, lx;
  91   double xhi;
  92
  93   xhi = ldbl_high (x);
  94   EXTRACT_WORDS (hx, lx, xhi);
  95   ix = hx & 0x7fffffff;
  96   if (ix < 0x3c600000)          /* x < 2**-57 */
  97     {
  98       if ((int) x == 0)         /* generate inexact */
  99         {
 100           if ((ix | lx | (iy + 1)) == 0)
 101             return one / fabs (x);
 102           else if (iy == 1)
 103             {
 104               if (fabsl (x) < LDBL_MIN)
 105                 {
 106                   long double force_underflow = x * x;
 107                   math_force_eval (force_underflow);
 108                 }
 109               return x;
 110             }
 111           else
 112             return -one / x;
 113         }
 114     }
 115   if (ix >= 0x3fe59420) /* |x| >= 0.6743316650390625 */
 116     {
 117       if ((hx & 0x80000000) != 0)
 118         {
 119           x = -x;
 120           y = -y;
 121           sign = -1;
 122         }
 123       else
 124         sign = 1;
 125       z = pio4hi - x;
 126       w = pio4lo - y;
 127       x = z + w;
 128       y = 0.0;
 129     }
 130   z = x * x;
 131   r = T0 + z * (T1 + z * (T2 + z * (T3 + z * T4)));
 132   v = U0 + z * (U1 + z * (U2 + z * (U3 + z * (U4 + z))));
 133   r = r / v;
 134
 135   s = z * x;
 136   r = y + z * (s * r + y);
 137   r += TH * s;
 138   w = x + r;
 139   if (ix >= 0x3fe59420)
 140     {
 141       v = (long double) iy;
 142       w = (v - 2.0 * (x - (w * w / (w + v) - r)));
 143       /* SIGN is set for arguments that reach this code, but not
 144          otherwise, resulting in warnings that it may be used
 145          uninitialized although in the cases where it is used it has
 146          always been set.  */
 147       DIAG_PUSH_NEEDS_COMMENT;
 148 #if __GNUC_PREREQ (4, 7)
 149       DIAG_IGNORE_NEEDS_COMMENT (5, "-Wmaybe-uninitialized");
 150 #else
 151       DIAG_IGNORE_NEEDS_COMMENT (5, "-Wuninitialized");
 152 #endif
 153       if (sign < 0)
 154         w = -w;
 155       DIAG_POP_NEEDS_COMMENT;
 156       return w;
 157     }
 158   if (iy == 1)
 159     return w;
 160   else
 161     {                           /* if allow error up to 2 ulp,
 162                                    simply return -1.0/(x+r) here */
 163       /*  compute -1.0/(x+r) accurately */
 164       long double u1, z1;
 165
 166       u1 = ldbl_high (w);
 167       v = r - (u1 - x);         /* u1+v = r+x */
 168       z = -1.0 / w;
 169       z1 = ldbl_high (z);
 170       s = 1.0 + z1 * u1;
 171       return z1 + z * (s + z1 * v);
 172     }
 173 }